Kramers-kronig receiver

ABSTRACT

There is provided a Kramers-Kronig receiver, comprising a reception path; wherein the reception path comprises: a Stokes receiver that is configured to receive a polarization-multiplexed signal and to output a Stokes vector; wherein the polarization-multiplexed signal comprises a first modulated signal, a second modulated signal and a continuous wave signal; wherein the first modulated signal is of a first polarization; wherein the second modulated signal is of a second polarization; wherein the continuous wave signal is of the first modulation or of the second modulation; a set of analog to digital converters that are configured to generate a digital representation of the Stokes vector; and a digital processor that is configured to process the digital representation of the Stokes vector to provide a reconstructed polarization-multiplexed signal, wherein the processing is based on a Kramers-Kronig relationship related to the polarization-multiplexed signal.

CROSS REFERENCE

This application claims priority from U.S. provisional patent Ser. No.62/347105 filing date Jun. 8, 2016 and from U.S. provisional patent Ser.No. 62/470436, filing date Mar. 13, 2017. Both US provisional patentsare incorporated herein in their entirety.

BACKGROUND

The interest for short-reach links of the kind needed for inter-datacenter communications has fueled in recent years the search fortransmission schemes that are simultaneously highly performing and costeffective.

Coherent optical transmission schemes are optimal from the standpoint ofspectral efficiency as they allow the encoding of information in bothquadratures and polarizations of the electric field. However, while theyconstitute the solution of choice for medium-to-long reach applications,the cost of a coherent receiver is a major obstacle in the case ofshort-reach links, whose role in many areas of applications is becomingincreasingly important. Indeed, coherent receivers used today are basedon the intradyne scheme, which requires two optical hybrids and fourpairs of balanced photodiodes, making its overall cost unacceptably highfor short links, such as those intended for inter-data centercommunications.

SUMMARY

There may be provided a Kramers-Kronig receiver that may include areception path. The reception path may include (i) a photodiode that maybe configured to receive a received signal and output a photocurrentthat represents the received signal; wherein the received signal mayinclude a continuous wave (CW) signal and a modulated signal; wherein afrequency gap between the CW signal and the modulated signal may besmaller than a bandwidth of the modulated signal; (ii) an analog todigital converter that may be configured to generate a digitalrepresentation of the photocurrent; and (iii) a digital processor thatmay be configured to process the digital representation of thephotocurrent to provide a reconstructed modulated signal, wherein theprocessing may be based on a Kramers-Kronig relationship related to thereceived signal.

The Kramers-Kronig relationship related to the received signal may be arelationship between a phase and an amplitude of the field of thereceived signal.

The digital processor may be configured to up-sample the digitalrepresentation of the photocurrent to provide an up-sampled digitalsignal.

The digital processor may be configured to calculate a logarithm of theup-sampled digital signal.

The digital processor may be configured to apply a Hilbert transform onthe logarithm of the up-sampled digital signal to provide a Hilberttransformed signal.

The digital processor may be configured to calculate a reconstructedphase of the received signal based on the Hilbert transformed signal.

The digital processor may be configured to calculate the reconstructedmodulated signal based on the reconstructed phase of the receivedsignal.

The reception path may consist essentially of the photodiode, the analogto digital converter and the digital processor.

The reception path may consist of the photodiode, the analog to digitalconverter and the digital processor.

The bandwidth of the analog to digital converter may be not smaller thana bandwidth of the received signal and may be smaller than twice thebandwidth of the received signal.

The bandwidth of the analog to digital converter may be not smaller thantwice a bandwidth of the received signal.

The digital processor may be configured to process the digitalrepresentation of the photocurrent without performing a logarithmoperation.

The Kramers-Kronig relationship related to the received signal may be arelationship between a real part and an imaginary part of a frequencyshifted version of the field of the received signal.

The digital processor may be configured to calculate in an iterativemanner the real part and the imaginary part of the frequency shiftedversion of the field of the received signal.

The digital processor may be configured to process the digitalrepresentation of the photocurrent without up-sampling.

The reception path does not include a local oscillator.

The reception path may include a local oscillator that may be configuredto output the continuous wave signal; and an adder that may beconfigured to add the continuous wave signal to the modulated signal toprovide the received signal.

There may be provided a Kramers-Kronig receiver that may include areception path; wherein the reception path may include (i) apolarization demultiplexing circuit that may be configured to receive apolarization-multiplexed signal and to output a first polarizationcomponent of the polarization-multiplexed signal and a secondpolarization component of the polarization-multiplexed signal; (ii) afirst photodiode that may be configured to receive a first intermediatesignal that may include a first continuous wave signal and the firstpolarization component, and output a first photocurrent that representsthe first intermediate signal; wherein a first frequency gap between thefirst continuous wave and the first intermediate signal does not exceeda bandwidth of the first intermediate signal; (ii) a second photodiodethat may be configured to receive a second intermediate signal that mayinclude a second continuous wave signal and the second polarizationcomponent, and output a second photocurrent that represents the secondintermediate signal; wherein a second frequency gap between the secondcontinuous wave and the second intermediate signal does not exceed abandwidth of the second intermediate signal; (iii) a first analog todigital converter that may be configured to generate a digitalrepresentation of the first photocurrent; (iv) a second analog todigital converter that may be configured to generate a digitalrepresentation of the second photocurrent; and (v) a digital processorthat may be configured to process the digital representation of thefirst photocurrent and the digital representation of the secondphotocurrent to provide a reconstructed polarization-multiplexed signal,wherein the processing may be based on one or more Kramers-Kronigrelationships related to the polarization-multiplexed signal.

The one or more Kramers-Kronig relationships related to thepolarization-multiplexed signal may include a Kramers-Kronigrelationship related to the first polarization component and aKramers-Kronig relationship related to the second polarizationcomponent.

The polarization demultiplexing circuit may include (a) a first combinerfor generating the first intermediate signal by combining the firstcontinuous wave signal and the first polarization component, and (b) asecond combiner for generating the second intermediate signal bycombining the second continuous wave signal and the second polarizationcomponent.

There may be provided a Kramers-Kronig receiver that may include areception path; wherein the reception path may include (i) a Stokesreceiver that may be configured to receive a polarization-multiplexedsignal and to output a Stokes vector; wherein thepolarization-multiplexed signal may include a first modulated signal, asecond modulated signal and a continuous wave signal; wherein the firstmodulated signal may be of a first polarization; wherein the secondmodulated signal may be of a second polarization; wherein the continuouswave signal may be of the first modulation or of the second modulation;(ii) a set of analog to digital converters that may be configured togenerate a digital representation of the Stokes vector; and (iii) adigital processor that may be configured to process the digitalrepresentation of the Stokes vector to provide a reconstructedpolarization-multiplexed signal, wherein the processing may be based ona Kramers-Kronig relationship related to the polarization-multiplexedsignal.

The polarization-multiplexed signal has a first polarization field and asecond polarization field; wherein the Stokes vector may include a firstStokes receiver output signal that may be indicative of (a) a square ofan absolute value of the first polarization field minus (b) a square ofan absolute value of the second polarization field.

The Stokes vector may include a second Stokes receiver output signalthat may be indicative of a real part of a given product of amultiplication of the first polarization field by the secondpolarization field, and a third Stokes receiver output signal that maybe indicative of an imaginary part of the given product.

The digital processor may be configured to (i) calculate a rotationmatrix that once applied to a time averaged Stokes vector provides arotated vector that has a first non-zero element, a zero valued secondelement and a zero valued third element; (ii) apply the rotation matrixon the Stokes vector to provide a rotated Stokes vector; and (iii)calculate the reconstructed polarization-multiplexed signal based on therotated Stokes vector.

There may be provided a Kramers-Kronig receiver that may include areception path, a transmission path and a local oscillator. The localoscillator may be configured to generate a continuous wave (CW) signal.The transmission path may include a complex modulator that may beconfigured to receive the CW signal and may be configured to modulatethe CW signal to generate a transmitted signal that may be transmittedto a communication link. The reception path may include (i) a combinerthat may be configured to add the CW signal with a received signal fromthe communication link to provide a combined signal; wherein thereceived signal may include a first sideband and a second sideband thatmay be spaced apart from each other by a guard band; wherein the CWsignal has a frequency that may be included in the guard band; (ii) afirst optical filter that may be configured to filter the combinedsignal and output a first intermediate signal that may include the CWsignal and the first sideband; (iii) a second optical filter that may beconfigured to filter the combined signal and output a secondintermediate signal that may include the CW signal and the secondsideband; (iv) a first photodiode that may be configured to receive thefirst intermediate signal and output a first photocurrent thatrepresents the first intermediate signal; (v) a second photodiode thatmay be configured to receive the second intermediate signal and output asecond photocurrent that represents the second intermediate signal; (vi)a first analog to digital converter that may be configured to generate adigital representation of the first photocurrent; (vii) a second analogto digital converter that may be configured to generate a digitalrepresentation of the second photocurrent; and (viii) a digitalprocessor that may be configured to process the digital representationof the first photocurrent and the digital representation of the secondphotocurrent to provide a reconstructed received signal, wherein theprocessing may be based on a Kramers-Kronig relationship related to thereceived signal.

The guard band may or may not exceed a portion of a bandwidth of thefirst sideband.

There may be provided a Kramers-Kronig receiver that may include areception path and a transmission path and a local oscillator that maybe configured to generate a continuous wave (CW) signal. Thetransmission path may include a complex modulator that may be configuredto receive the CW signal and may be configured to modulate the CW signalto generate a transmitted signal that may be transmitted to acommunication link. The reception path may include (i) a polarizationdemultiplexing circuit that may be configured to receive apolarization-multiplexed signal and to output a first polarizationcomponent of the polarization-multiplexed signal and a secondpolarization component of the polarization-multiplexed signal; (ii) afirst optical filter that may be configured to filter the firstpolarization component and output a first sideband of the firstpolarization component; (iii) a second optical filter that may beconfigured to filter the first polarization component and output asecond sideband of the first polarization component; wherein the firstand second sidebands of the first polarization component may be spacedapart from each other by a guard band; (iv) a first combiner that may beconfigured to add the CW signal with the first sideband of the firstpolarization component to provide a first combined signal; (v) a secondcombiner that may be configured to add the CW signal with the secondsideband of the first polarization component to provide a secondcombined signal; (vi) a first photodiode that may be configured toreceive the first combined signal and output a first photocurrent thatrepresents the first combined signal; (vii) a second photodiode that maybe configured to receive the second combined signal and output a secondphotocurrent that represents the second combined signal; (viii) a firstanalog to digital converter that may be configured to generate a digitalrepresentation of the first photocurrent; (ix) a second analog todigital converter that may be configured to generate a digitalrepresentation of the second photocurrent; (x) a third optical filterthat may be configured to filter the second polarization component andoutput a first sideband of the second polarization component; (xi) afourth optical filter that may be configured to filter the secondpolarization component and output a second sideband of the secondpolarization component; wherein the first and second sidebands of thesecond polarization component may be spaced apart from each other by theguard band; (xii) a third combiner that may be configured to add the CWsignal with the first sideband of the second polarization component toprovide a third combined signal; (xiii) a fourth combiner that may beconfigured to add the CW signal with the second sideband of the secondpolarization component to provide a fourth combined signal; (xiv) athird photodiode that may be configured to receive the third combinedsignal and output a third photocurrent that represents the thirdcombined signal; (xv) a fourth photodiode that may be configured toreceive the fourth combined signal and output a fourth photocurrent thatrepresents the fourth combined signal; (xvi) a third analog to digitalconverter that may be configured to generate a digital representation ofthe third photocurrent; (xvii) a fourth analog to digital converter thatmay be configured to generate a digital representation of the fourthphotocurrent; and (xviii) a digital processor that may be configured toprocess the digital representation of the first photocurrent, the secondphotocurrent, the third photocurrent and the fourth photocurrent toprovide a reconstructed polarization-multiplexed signal, wherein theprocessing may be based on a Kramers-Kronig relationship related to thepolarization-multiplexed signal.

There may be provided a method for receiving and processing signalsusing any of the Kramers-Kronig receivers illustrated above.

The method may include receiving a signal and reconstructing the signalbased on at least one Kramers-Kronig relationship related to the signal.

There may be provided a method for receiving and reconstructing areceived signal. The method may include (i) receiving, by a photodiodeof a reception path of Kramers-Kronig receiver, a received signal; (ii)outputting by the photodiode a photocurrent that represents the receivedsignal; wherein the received signal may include a continuous wave (CW)signal and a modulated signal; wherein a frequency gap between the CWsignal and the modulated signal may be smaller than a bandwidth of themodulated signal; (iii) performing an analog to digital conversion ofthe photocurrent, by an analog to digital converter of the receptionpath, to generate a digital representation of the photocurrent; and(iii) processing, by a digital processor of the reception path, thedigital representation of the photocurrent to provide a reconstructedmodulated signal, wherein the processing may be based on aKramers-Kronig relationship related to the received signal.

The Kramers-Kronig relationship related to the received signal may be arelationship between a phase and an amplitude of the field of thereceived signal.

The processing may include up-sampling the digital representation of thephotocurrent to provide an up-sampled digital signal.

The processing may include calculating a logarithm of the up-sampleddigital signal.

The processing may include applying a Hilbert transform on the logarithmof the up-sampled digital signal to provide a Hilbert transformedsignal.

The processing may include calculating a reconstructed phase of thereceived signal based on the Hilbert transformed signal.

The processing may include calculating the reconstructed modulatedsignal based on the reconstructed phase of the received signal.

The reception path may consist essentially of the photodiode, the analogto digital converter and the digital processor.

The reception path may consist of the photodiode, the analog to digitalconverter and the digital processor.

The bandwidth of the analog to digital converter may be not smaller thana bandwidth of the received signal and may be smaller than twice thebandwidth of the received signal.

The bandwidth of the analog to digital converter may be not smaller thantwice a bandwidth of the received signal.

The processing may exclude performing a logarithm operation.

The Kramers-Kronig relationship related to the received signal may be arelationship between a real part and an imaginary part of a frequencyshifted version of the field of the received signal.

The processing may include calculating in an iterative manner the realpart and the imaginary part of the frequency shifted version of thefield of the received signal.

The processing may exclude up-sampling.

The reception path may or may not include a local oscillator.

The method may include outputting by a local oscillator of theKramers-Kronig receiver the continuous wave signal; and adding thecontinuous wave signal to the modulated signal to provide the receivedsignal.

There may be provided a method for receiving and reconstructing apolarization-multiplexed signal. The method may include (i) receiving bythe polarization demultiplexing circuit of a reception path ofKramers-Kronig receiver, a polarization-multiplexed signal; (ii)outputting by the polarization demultiplexing circuit a firstpolarization component of the polarization-multiplexed signal and asecond polarization component of the polarization-multiplexed signal;(iii) receiving by a first photodiode of the reception path ofKramers-Kronig receiver, a first intermediate signal that may include afirst continuous wave signal and the first polarization component, (iv)outputting by the first photodiode a first photocurrent that representsthe first intermediate signal; wherein a first frequency gap between thefirst continuous wave and the first intermediate signal does not exceeda bandwidth of the first intermediate signal; (v) receiving by a secondphotodiode of the reception path of Kramers-Kronig receiver a secondintermediate signal that may include a second continuous wave signal andthe second polarization component, (vi) outputting by the secondphotodiode a second photocurrent that represents the second intermediatesignal; wherein a second frequency gap between the second continuouswave and the second intermediate signal does not exceed a bandwidth ofthe second intermediate signal; (vii) performing an analog to digitalconversion of the first photocurrent, by a first analog to digitalconverter of the reception path, to generate a digital representation ofthe first photocurrent; (viii) performing an analog to digitalconversion of the second photocurrent, by a second analog to digitalconverter of the reception path, to generate a digital representation ofthe second photocurrent; (ix) processing, by a digital processor of thereception path the digital representation of the first photocurrent andthe digital representation of the second photocurrent to provide areconstructed polarization-multiplexed signal, wherein the processingmay be based on one or more Kramers-Kronig relationships related to thepolarization-multiplexed signal.

The one or more Kramers-Kronig relationships related to thepolarization-multiplexed signal may include a Kramers-Kronigrelationship related to the first polarization component and aKramers-Kronig relationship related to the second polarizationcomponent.

The method may include generating, by a first combiner the firstintermediate signal by combining the first continuous wave signal andthe first polarization component, and (b) a second combiner forgenerating the second intermediate signal by combining the secondcontinuous wave signal and the second polarization component.

There may be provided a method for receiving and reconstructing apolarization-multiplexed signal, the method may include (i) receiving,by a Stokes receiver of a reception path of Kramers-Kronig receiver, thepolarization-multiplexed signal; (ii) outputting by the Stokes receiver,a Stokes vector; wherein the polarization-multiplexed signal may includea first modulated signal, a second modulated signal and a continuouswave signal; wherein the first modulated signal may be of a firstpolarization; wherein the second modulated signal may be of a secondpolarization; wherein the continuous wave signal may be of the firstmodulation or of the second modulation; (iii) performing an analog todigital conversion, by a set of analog to digital converters, of theStokes vector to provide a digital representation of the Stokes vector;and (iv) processing, by a digital processor of the reception path ofKramers-Kronig receiver, the digital representation of the Stokes vectorto provide a reconstructed polarization-multiplexed signal, wherein theprocessing may be based on a Kramers-Kronig relationship related to thepolarization-multiplexed signal.

The polarization-multiplexed signal has a first polarization field and asecond polarization field; wherein the Stokes vector may include a firstStokes receiver output signal that may be indicative of (a) a square ofan absolute value of the first polarization field minus (b) a square ofan absolute value of the second polarization field.

The Stokes vector may include a second Stokes receiver output signalthat may be indicative of a real part of a given product of amultiplication of the first polarization field by the secondpolarization field, and a third Stokes receiver output signal that maybe indicative of an imaginary part of the given product.

The processing may include (i) calculating a rotation matrix that onceapplied to a time averaged Stokes vector provides a rotated vector thathas a first non-zero element, a zero valued second element and a zerovalued third element; (ii) applying the rotation matrix on the Stokesvector to provide a rotated Stokes vector; and (iii) calculating thereconstructed polarization-multiplexed signal based on the rotatedStokes vector.

There may be provided a method. The method may include (i) generating bya local oscillator a continuous wave (CW) signal; (ii) modulating, by acomplex modulator, the CW signal to generate a transmitted signal thatmay be transmitted to a communication link, (iii) adding, by a combinerof a reception path of Kramers-Kronig receiver, the CW signal with areceived signal from the communication link to provide a combinedsignal; wherein the received signal may include a first sideband and asecond sideband that may be spaced apart from each other by a guardband; wherein the CW signal has a frequency that may be included in theguard band; (iv) filtering, by a first optical filter of the receptionpath of Kramers-Kronig receiver, the combined signal and output a firstintermediate signal that may include the CW signal and the firstsideband; (v) filtering, by a second optical filter of the receptionpath of Kramers-Kronig receiver, the combined signal and output a secondintermediate signal that may include the CW signal and the secondsideband; (vi) receiving by a first photodiode of the reception path ofKramers-Kronig receiver the first intermediate signal and outputting afirst photocurrent that represents the first intermediate signal; (vii)receiving, by a second photodiode of the reception path ofKramers-Kronig receiver, the second intermediate signal and outputting asecond photocurrent that represents the second intermediate signal;(viii) performing an analog to digital conversion of the firstphotocurrent, by a first analog to digital converter of the receptionpath, to generate a digital representation of the first photocurrent;(ix) performing an analog to digital conversion of the secondphotocurrent, by a second analog to digital converter of the receptionpath, to generate a digital representation of the second photocurrent;and (x) processing, by a digital processor of the reception path ofKramers-Kronig receiver, the digital representation of the firstphotocurrent and the digital representation of the second photocurrentto provide a reconstructed received signal, wherein the processing maybe based on a Kramers-Kronig relationship related to the receivedsignal.

The guard band may or may not exceed a portion of a bandwidth of thefirst sideband.

There may be provided a method. The method may include (i) generating bya local oscillator a continuous wave (CW) signal; (ii) modulating, by acomplex modulator, the CW signal to generate a transmitted signal thatmay be transmitted to a communication link, (iii) receiving apolarization-multiplexed signal by a polarization demultiplexing circuitof the reception path of Kramers-Kronig receiver and outputting a firstpolarization component of the polarization-multiplexed signal and asecond polarization component of the polarization-multiplexed signal;(iv) filtering, by a first optical filter of the reception path ofKramers-Kronig receiver, the first polarization component and outputtinga first sideband of the first polarization component; (v) filtering, bya second optical filter of the reception path of Kramers-Kronigreceiver, the first polarization component and outputting a secondsideband of the first polarization component; wherein the first andsecond sidebands of the first polarization component may be spaced apartfrom each other by a guard band; (vi) combining, by a first combiner ofthe reception path of Kramers-Kronig receiver, the CW signal with thefirst sideband of the first polarization component to provide a firstcombined signal; (vii) combining, by a second combiner of the receptionpath of Kramers-Kronig receiver the CW signal with the second sidebandof the first polarization component to provide a second combined signal;(vii) receiving, by a first photodiode of the reception path ofKramers-Kronig receiver, the first combined signal and outputting afirst photocurrent that represents the first combined signal; (viii)receiving, by a second photodiode of the reception path ofKramers-Kronig receiver, the second combined signal and outputting asecond photocurrent that represents the second combined signal; (ix)converting, by a first analog to digital converter of the reception pathof Kramers-Kronig receiver, the first photocurrent to provide a digitalrepresentation of the first photocurrent; (x) converting, by a secondanalog to digital converter of the reception path of Kramers-Kronigreceiver, the second photocurrent to provide a digital representation ofthe second photocurrent; (xi) filtering, by a third optical filter ofthe reception path of Kramers-Kronig receiver, the second polarizationcomponent and outputting a first sideband of the second polarizationcomponent; (xii) filtering, by a fourth optical filter of the receptionpath of Kramers-Kronig receiver, the second polarization component andoutputting a second sideband of the second polarization component;wherein the first and second sidebands of the second polarizationcomponent may be spaced apart from each other by a guard band; (xiii)combining, by a third combiner of the reception path of Kramers-Kronigreceiver, the CW signal with the first sideband of the secondpolarization component to provide a third combined signal; (xiv)combining, by a fourth combiner of the reception path of Kramers-Kronigreceiver the CW signal with the second sideband of the secondpolarization component to provide a fourth combined signal; (xv)receiving, by a third photodiode of the reception path of Kramers-Kronigreceiver, the third combined signal and outputting a third photocurrentthat represents the third combined signal; (xvi) receiving, by a fourthphotodiode of the reception path of Kramers-Kronig receiver, the fourthcombined signal and outputting a fourth photocurrent that represents thefourth combined signal; (xvii) converting, by a third analog to digitalconverter of the reception path of Kramers-Kronig receiver, the thirdphotocurrent to provide a digital representation of the thirdphotocurrent; (xviii) converting, by a fourth analog to digitalconverter of the reception path of Kramers-Kronig receiver, the fourthphotocurrent to provide a digital representation of the fourthphotocurrent; (xix) processing, by a digital processor of the receptionpath the digital representation of the first photocurrent, the secondphotocurrent, the third photocurrent and the fourth photocurrent toprovide a reconstructed polarization-multiplexed signal, wherein theprocessing may be based on a Kramers-Kronig relationship related to thepolarization-multiplexed signal.

BRIEF DESCRIPTION OF THE DRAWINGS

The subject matter regarded as the invention is particularly pointed outand distinctly claimed in the concluding portion of the specification.The invention, however, both as to organization and method of operation,together with objects, features, and advantages thereof, may best beunderstood by reference to the following detailed description when readwith the accompanying drawings in which:

FIG. 1A illustrates a Kramers-Kronig receiver according to at least oneembodiment of the invention;

FIG. 1B illustrates a Kramers-Kronig receiver according to at least oneembodiment of the invention;

FIG. 2 illustrates absolute values of an original waveform and areconstructed waveform according to at least one embodiment of theinvention;

FIG. 3 illustrates received constellations according to at least oneembodiment of the invention;

FIG. 4 illustrates a bit error rate (BER) versus optical signal to noiseratio (OSNR) for a 24 Gbaud 16 quadrature amplitude modulation (QAM)modulated signal according to at least one embodiment of the invention;

FIG. 5 illustrates BER versus total transmit power according to at leastone embodiment of the invention;

FIG. 6 illustrates a Kramers-Kronig receiver according to at least oneembodiment of the invention;

FIG. 7A illustrates a Stokes Kramers-Kronig receiver according to atleast one embodiment of the invention;

FIG. 7B illustrates a Kramers-Kronig receiver according to at least oneembodiment of the invention;

FIG. 8 illustrates a BER versus OSNR according to at least oneembodiment of the invention;

FIG. 9A illustrates a BER versus OSNR according to at least oneembodiment of the invention;

FIG. 9B illustrates a BER versus transmitted power according to at leastone embodiment of the invention;

FIG. 10 illustrates a BER versus OSNR according to at least oneembodiment of the invention;

FIG. 11 illustrates a transceiver according to at least one embodimentof the invention;

FIG. 12 illustrates a transceiver according to at least one embodimentof the invention;

FIG. 13 illustrates a transceiver according to at least one embodimentof the invention; and

FIG. 14 illustrates a method according to at least one embodiment of theinvention.

DETAILED DESCTIPRION OF THE DRAWINGS

In the following detailed description, numerous specific details are setforth in order to provide a thorough understanding of the invention.However, it will be understood by those skilled in the art that thepresent invention may be practiced without these specific details. Inother instances, well-known methods, procedures, and components have notbeen described in detail so as not to obscure the present invention.

The subject matter regarded as the invention is particularly pointed outand distinctly claimed in the concluding portion of the specification.The invention, however, both as to organization and method of operation,together with objects, features, and advantages thereof, may best beunderstood by reference to the following detailed description when readwith the accompanying drawings.

It will be appreciated that for simplicity and clarity of illustration,elements shown in the figures have not necessarily been drawn to scale.For example, the dimensions of some of the elements may be exaggeratedrelative to other elements for clarity. Further, where consideredappropriate, reference numerals may be repeated among the figures toindicate corresponding or analogous elements.

Because the illustrated embodiments of the present invention may for themost part, be implemented using electronic components and circuits knownto those skilled in the art, details will not be explained in anygreater extent than that considered necessary as illustrated above, forthe understanding and appreciation of the underlying concepts of thepresent invention and in order not to obfuscate or distract from theteachings of the present invention.

Any reference in the specification to a method should be applied mutatismutandis to a system capable of executing the method and should beapplied mutatis mutandis to a non-transitory computer readable mediumthat stores instructions that once executed by a computer result in theexecution of the method.

Any reference in the specification to a system should be applied mutatismutandis to a method that may be executed by the system and should beapplied mutatis mutandis to a non-transitory computer readable mediumthat stores instructions that may be executed by the system.

Any reference in the specification to a non-transitory computer readablemedium should be applied mutatis mutandis to a system capable ofexecuting the instructions stored in the non-transitory computerreadable medium and should be applied mutatis mutandis to method thatmay be executed by a computer that reads the instructions stored in thenon-transitory computer readable medium.

The term “field” refers to an electric field.

There is provided a direct-detection coherent receiver which combinesthe advantages of coherent transmission and the cost-effectiveness ofdirect detection. The working principle of the proposed receiver isbased on the Kramers-Kronig relations, and its implementation requireshaving a continuous-wave signal at one edge of the information-carryingsignal spectrum.

The Kramers-Kronig receiver scheme allows digital post-compensation oflinear propagation impairments and, as compared to other existingsolutions, is more efficient in terms of spectral occupancy and energyconsumption.

The Kramers-Kronig receiver may include hardware components such as aradio frequency (RF) front end or interface, one or more antennas,frequency downconverters, adders, digital and/or analog filters, memory,and a hardware processor. The hardware processor may executeinstructions for digital processing.

The suggested scheme eliminates the need for a frequency gap between thelocal oscillator (continuous wave signal) and the modulated signal (alsoreferred to as information carrying signal) and hence increases thespectral efficiency by a factor of two in comparison with theself-heterodyne scheme illustrated in reference [1]—B. J. C. Schmidt, A.J. Lowery, and J. Armstrong, “Experimental Demonstrations of ElectronicDispersion Compensation for Long-Haul Transmission UsingDirect-Detection Optical OFDM,” J. Lightwave Technol. 26, 196-203(2008). The self-heterodyne scheme requires a gap of a magnitude of thebandwidth of the modulated signal.

The continuous wave signal may be provided by the transmitter (includedin a received signal) or may be added by a local oscillator of thereceiver.

The extraction of the received field from the photocurrent is performeddigitally while taking advantage of the Kramers-Kronig relationship—suchas the Kramers-Kronig relationship between the phase and amplitude ofthe field of the received signal—the field that impinges upon thephotodiode. Yet another Kramers-Kronig relationship is the relationshipbetween the real and imaginary parts of the received signal.

The Kramers-Kronig relations are somewhat ubiquitous, as they emerge invarious areas of physics and engineering. Their applicability in thecontext of direct-detection coherent receivers follows from a simpleproperty of minimum phase signals.

The proposed scheme allows the full reconstruction of the complexenvelope of the filed impinging upon the photodiode, and hence it iscompatible with DSP-based digital compensation of propagation-inducedlinear impairments.

The gain in spectral efficiency may come at the expense of a somewhatmore stringent requirement on the power of the CW signal in comparisonwith the self-heterodyne scheme. Nonetheless, it is shown in whatfollows, in practical scenarios this power requirement can beconsiderably relaxed.

The field reconstruction procedure performed by the Kramers-Kronigreceiver is illustrated below.

A sufficient condition for distortion-free operation of theKramers-Kronig receiver implies that if s(t) is a complex signal whosespectrum is contained between −B/2 and B/2, thenh(t)=A+s(t)exp^((−iπBt)) is a minimum phase signal, provided that|s(t)|<A, for all t. In this case, denoting h(t)=|h(t)|exp ^([iφ(t))],the absolute value |h(t)| and the argument φ(t) are uniquely related toeach other via the Hilbert transform:

$\begin{matrix}{{{\varphi (t)} = {\frac{1}{\pi}{p.v.{\int_{- \infty}^{\infty}{{dt}^{\prime}\frac{\log \left\lbrack {{h\left( t^{\prime} \right)}} \right\rbrack}{t - t^{\prime}}}}}}},} & (1)\end{matrix}$

Where p.v. stands for principal value. Equation (1) is a Kramers-Kronigrelation existing between φ and log [|h|], and is most convenientlyimplemented in the frequency domain where it assumes the form

{tilde over (ϕ)}(ω)=i sing(ω)

{log[|h(t)|]},  (2)

Where sign(ω) is the sign function, which is equal to 1 for ω>0, to 0for ω=0, and to −1 for ω<0, and where F denotes a Fourier transform. Itshould be noted that the phase reconstruction is exact up to a constantphase offset. This can be seen most easily from Equation (2), where thezero-frequency component of {tilde over (φ)}(ω) is set to zero by thesign function.

We now proceed to describing the field reconstruction procedureperformed by the Kramers-Kronig receiver, where we assume for simplicitya scalar description of the field that adequately represents the case inwhich no polarization multiplexing is done.

Schemes that accommodate polarization multiplexing will be introducedsubsequently We denote the complex envelope of the field (of thereceived signal) by E_(s)(t) which is assumed to be contained within afinite optical bandwidth denoted by B. The local oscillator is assumedto output a continuous wave (CW) signal whose amplitude is E₀ and whosefrequency may coincide with the left edge (or the right edge) of theinformation-carrying signal spectrum.

Some of the following formulas refer to a case where the CW signal islocated to the left (lower frequency) of the modulated signal. Theseequations are applicable mutatis mutandis to a case where the CW signalis located to the right (higher frequency) of the modulated signal.

We assume that E₀ is real-valued and positive, which is equivalent toreferring all phase values to that of the local oscillator. The complexenvelope of the field impinging upon the photodiode is thusE(t)=E_(s)(t)+E₀ exp(iπBt).

Provided that the local oscillator amplitude is chosen such thatE₀>|E_(s)(t)| for all t, the signal E(t)e^(−iπBt)=E₀+E_(s)(t)e^(−πBt) isguaranteed to be minimum phase. In practice, E(t) can be minimum phaseeven if the condition E₀>|E_(s)(t)| is not satisfied for all values oft. A less restrictive condition is that the trajectories of E(t) in thecomplex plane do not encircle the origin.

Thus, the signal E(t) can be appropriate for our form of communicationseven when the condition E₀>|E_(s)(t)| is not explicitly satisfied.Hence, the determination of the appropriate minimum value of E₀ can bedone on the basis of the less restrictive condition, or it can beperformed by empirical optimization.

The photocurrent I produced by the photodiode is proportional to thefield intensity I=|E(t)|², where we have set the proportionalitycoefficient to 1, for the sake of simplicity. Hence, using Equations (1)and (2), the signal E_(s)(t) can be reconstructed as follows

$\begin{matrix}{{{E_{s}(t)} = {\left\{ {{\sqrt{I(t)}{\exp \left\lbrack {i\; {\varphi_{E}(t)}} \right\rbrack}} - E_{0}} \right\} {\exp \left( {i\; \pi \; {Bt}} \right)}}},} & (3) \\{{\varphi_{E}(t)} = {\frac{1}{2\pi}{p.v.{\int_{- \infty}^{\infty}{{dt}^{\prime}{\frac{\log \left\lbrack {I\left( t^{\prime} \right)} \right\rbrack}{t - t^{\prime}}.}}}}}} & (4)\end{matrix}$

A possible issue with the signal reconstruction described above is thatthe logarithm appearing in Equation (4) introduces spectral broadeningwhich may necessitate digital up-sampling of the received photocurrent.

Another signal reconstruction approach that resolves this issue makesuse of the fact that the frequency shifted signal E′_(s)(t)=E_(s)(t)e^(−iπBt) has real and imaginary parts E′_(s,r)(t) andE′_(s,i)(t) satisfying the Kramers-Kronig relations,

$\begin{matrix}{{E_{s,i}^{\prime}(t)} = {\frac{1}{\pi}{p.v.{\int_{- \infty}^{\infty}{{dt}^{\prime}{\frac{E_{s,r}^{\prime}\left( t^{\prime} \right)}{t - t^{\prime}}.}}}}}} & (5)\end{matrix}$

Using I(t)=|E₀+E′_(s)(t)| (and referring to E₀ as real valued withoutloss of generality), one obtains the equality

I(t)=E′ ₀ ² +E _(s,r) ²(t)+E′ _(s,i) ²(t)+2E ₀ E′ _(s,r)(t)  (6)

Once E′_(s)(t) is replaced by its expression given by Equation (5),Equation (6) becomes an integral equation that, being equivalent toEquations (3) and (4), has a unique solution if E₀>|E_(S)(t)| for all t,which can be obtained by means of the iterative procedure described inwhat follows. Equation (6) can be formally solved for E′_(s,r)(t) withthe result

E′ _(s,r)(t)=√{square root over (I(t)−E′ _(s,i) ²(t))}−E ₀,  (7)

where we have taken the positive determination of the square rootbecause it is the only one consistent with the Kramers-Kronig condition,E₀>|E_(S)(t)|. Equation (7) can be solved by iterations, where in thefirst step one solves for E′_(s,r)(t) while setting E′_(s,i)(t)=0 on theright-hand side of (7). The resulting E′_(s,r)(t) is then filtered usinga square filter of bandwidth B, and used to extract the next iterationof E′_(s,i)(t) through Equation. (5).

The procedure is then repeated until the values of E′_(s,r)(t) andE′_(s,i)(t) stabilize to a desired extent. For demonstrating theKramers-Kronig scheme we assume that the oversampling of thephotocurrent is not an issue and hence in what follows we apply theprocedure represented in Equations (1)-(4).

The other signal reconstruction approach (part of the processing by thedigital processor—may also referred to a Kramers-Kronig algorithm) mayinclude:

-   -   a. Setting E_(s,i)=0 and find E_(s,r) from Equation (7).    -   b. Inserting E_(s,r) into Equation (5) find E_(s,i).    -   c. Filtering E_(s,i) in a filter bandwidth of B (passband        between 0 and B).    -   d. Using E_(s,i) to find E_(s,r) from Equation (7).    -   e. Filtering E_(s,r) in a filter bandwidth of B (passband        between 0 and B).    -   f. Repeating steps b, c, d and e for a predefined number of        times in order to obtain desired accuracy. Typically, three        repetitions (iterations) are sufficiently accurate.

Various implementations of the Kramers-Kronig receiver are plotted inFIGS. 1A, 1B and 6.

Referring to FIG. 1A—a received signal that includes a continuous wavesignal (CW signal at a left edge of the modulated signal) and amodulated signal 12 (of bandwidth B) is received by a reception path 20of the Kramers-Kronig receiver.

The received signal, is directly detected by photodiode 21, then sampledat the sampling rate of 2B (by analog to digital converter 30 that has abandwidth of at least 2B). The digital samples are finally processed (bydigital processor 40) for chromatic dispersion compensation andextraction of the transmitted symbols. The processing may includeremoving the CW signal, or removing an estimated CW signal, removing anaverage value (DC) of the received signal. The digital processor mayperform, for example phase recovery, chromatic dispersion compensation,and possibly nonlinear distortion compensation.

The digital processor 40 may performs a Kramers-Kronig algorithm byexecuting the steps of:

-   -   i. Up-sample the digital representation of the photocurrent I(t)    -   ii. Calculate a reconstructed phase φ_(E)(t)=0.5*Hilbert{log        (I(t)}.    -   iii. Calculate a reconstructed modulated signal        E_(s)(t)=√{square root over (I(t))}e^(iϕ) ^(E) ^((t))−E₀, where        E₀ is the complex envelope of the CW signal, which may be        evaluated as the time average of √{square root over        (I(t))}e^(iϕ) ^(E) ^((t))    -   iv. Perform coherent processing steps.

FIG. 1B shows an alternative implementation, where the local oscillator28 is added (by adder 29) to the modulated signal at the receiver usinga frequency-selective coupler to avoid signal and local oscillator loss.In this case the transmitter does not transmit the CW signal.

This implementation accommodates polarization multiplexing (asillustrated in FIG. 7B), eliminates the need for an optical hybrid atthe receiver, and requires the use of a single photodiode. However, itmay require a local oscillator laser.

Alternatively—the CW signal can be extracted from the received signal—incase were the transmitter also transmits a CW signal in addition to theinformation-carrying signal.

An alternative method for extracting the signal without performing thelog operation and hence avoiding the need for up-sampling was discussedin the previous section.

Numerical Validation of the Kramers-Kronig Receiver

We start by providing a numerical proof-of-concept of the Kramers-Kronigreceiver scheme. To that end we consider the case in which the signalimpinging upon the receiver is produced by filtering white Gaussiannoise with a square optical filter of bandwidth B. Since Gaussian noiseis characterized by the largest entropy that can be carried by signalsof the same bandwidth, it seems natural to use it for validating theKramers-Kronig receiver concept.

FIG. 2 shows the absolute values of the original and the reconstructedwaveforms by solid and dot-dashed lines, respectively (a similar picturecorresponds to the real or imaginary parts). The computation wasperformed with B=32 GHz, using a time window T=1024/B. The signalsamples used to perform the Hilbert transform were taken at a rate of 2B.

The signal used in the simulation was generated as a complex-valuedcircular Gaussian waveform with a flat-top spectrum of width B=32 GHz,which was normalized such that its largest absolute value within thesimulated time window was 0.99E₀. The top panel shows the signalintensity within the entire simulated time window, whereas the bottompanels zoom into the waveform in two distinct intervals. In the leftpanel, we show the beginning of the frame, where the edge effect of theHilbert transform is visible. The waveform reconstruction error is largeinitially, but then it reduces as the distance from the beginning of theframe increases. The waveform in the right panel is taken from themiddle of the frame, where the edge effect is absent, and the quality ofthe reconstruction is excellent. As can be seen in the figure, theduration of the edge effect is of the order of 0.5 ns, which isequivalent to ˜16/B. The edges in the beginning and in the end of theprocessed frame will have to be discarded in a practical implementationof the Kramers-Kronig receiver. As is discussed in what follows, asimilar situation, where the edges of the processed frame need to bediscarded, characterizes the digital compensation of chromaticdispersion.

Again—referring to FIG. 2—panel 110 shows the absolute values of theoriginal waveform (continuous) and reconstructed (dashed) waveforms.Panels 120 and 130 zoom into the beginning and the center of the frame,respectively.

Linear Transmission Performance

We now switch to demonstrating the implementation of the Kramers-Kronigtransceiver in the context of a digital communication systemtransmitting QAM signals. We focus first on the linear regime ofoperation, where the effect of the fiber nonlinearity is negligible.FIG. 3 refers to the back-to-back configuration. Back to backconfiguration describes a scenario in which the transmitter and thereceiver are connected directly with each other without an optical fiberof with an optical fiber of negligible length. or connected Panels 141,142, 143 and 144 show the received constellations for single-channel16-QAM signaling, where a raised cosine fundamental waveform with 0.05roll-off factor was assumed. Since this figure was plotted in the regimeof linear transmission, and in the absence of noise, the displayedresults are not affected by the baudrate.

The various panels correspond to different settings of the localoscillator power P_(LO)=E₀ ². In the leftmost panel (141) the localoscillator power was set to 1.1 times the maximum value of theinformation carrying signal power, corresponding to about 11 dB abovethe average signal power, and the received constellation was indeedperfect (the edge-effect seen in FIG. 2 was taken care of by discardingthe symbols at the edges of the simulation time window).

In the remaining three panels, the CW signal power level was reduced to8 db (142), 6 db (143) and 3 dB (144) above the average channel power.

FIG. 3 shows that the received constellation quality deteriorates as thepower of the CW signal is reduced. The scattering of the constellationpoints is caused by the fact that the instantaneous power of theinformation carrying signal occasionally exceeds the CW signal power,thus violating the minimum-phase condition for signal reconstructionunderpinning the Kramers-Kronig receiver concept.

Practical considerations suggest that one should pick the lowest localoscillator power for which the reconstruction noise shown in FIG. 3 issufficiently small to allow reliable detection. Indeed, large values ofthe local oscillator power would deteriorate the system's powerefficiency and, as we show in what follows, reduce its tolerance topropagation-induced nonlinear distortions.

FIG. 4 includes panel 150 and illustrates the penalty introduced by anoncompliant CW signal power in the presence of amplification noise.Plotted in the figure is the BER as a function of OSNR for a 24 Gbaud 16QAM modulated signal, for a range of LO powers. Empty markers show theresults obtained in the back-to-back configuration, whereas filledmarkers refer to the case of linear transmission through a 100 kmsingle-mode fiber link. In the latter case, chromatic dispersion wascompensated electronically at the receiver, after signal reconstruction.In all simulations, the information-carrying signal consisted of apseudo-random sequence of 2¹⁴ symbols. Prior to reception, amplificationnoise was added to the signal and then a 12th order super-Gaussianoptical filter with a 3 dB bandwidth of 40 GHz was applied. The figureconfirms the benefit of increasing the LO power: at low levels the BERsaturates for increasing OSNR, owing to the errors caused by imperfectsignal reconstruction. This saturation tends to disappear when the LOpower is sufficiently large.

The difference between the back-to-back results and the results obtainedwith SMF transmission shows that electronic CD compensation implies anOSNR penalty, as well as an increase in the smallest achievable BER. Thereason for this penalty is in the larger peak-to-average power ratio(PAPR) of the CD-impaired information carrying signal, as compared tothe PAPR of the launched signal. The OSNR penalty could in principle beavoided either by pre-compensating the modulated signal, or byimplementing optical CD compensation at the receiver, although thiswould imply an obvious complication of the transceiver structure.

FIG. 4 illustrates BER versus OSNR for a 24 Gbaud 16 QAM modulatedsignal. Each curve was obtained by setting the power of the localoscillator to the value shown in the legend. Empty markers refer to theback-to-back configuration, while filled markers were obtained for a 100km single-span link, where CD was compensated electronically at thereceiver after signal reconstruction.

The results of FIG. 4 indicate that for OSNR values higher than 20 dB,pre-FEC BERs lower than 10⁻² can be achieved with a local oscillatorpower exceeding the average channel power by about 7 dB (implying thatthe total transmitted power increases by 7.8 dB relative to coherenttransmission). This makes the Kramers-Kronig scheme considerably morepower efficient than the scheme of reference [2]—S. Randel, D. Pilori,S. Chandrasekhar, G. Raybon, and P. J. Winzer “100-Gb/sDiscrete-Multitone Transmission Over 80-km SSMF Using Single-SidebandModulation with Novel Interference-Cancellation Scheme,” Proc. ofEuropean Conference of Optical Communications 2015 (ECOC15),Valencia—Spain, Paper 0697 (2015).

On the other hand, it should be stressed that the Kramers-Kronig schemeis twice more spectrally efficient than IMDD—the lower spectralefficiency of IMDD follows from the fact that with intensity modulationno information is encoded into the optical phase. This makes IMDDinferior even to single-quadrature modulation, where positive andnegative amplitude values can be used) and the self-heterodyne scheme.

A detailed comparison with other known direct-detection schemes in termsof spectral efficiency, power efficiency, and amenability to digitaldispersion compensation, is shown in Table 1. To facilitate thecomparison of spectral efficiencies, we express the optical bandwidth interms of R, which is the lowest sampling rate that allows reconstructionof the detected photocurrent.

As can be seen from the table, the Kramers-Kronig scheme provides anattractive combination of properties.

Table 1 includes a comparison between various schemes in terms ofoptical bandwidth, suitability for digital compensation of linearimpairments, and power efficiency. The symbol R represents the bandwidthof the detected photocurrent, which is also the lowest acceptablesampling rate.

The ratio P_(LO)/P_(S) ranges from 10 to 20 in reference [3]—M.Schuster, S. Randel, C. A. Bunge, S. C. J. Lee, F. Breyer, B. Spinnler,and K. Petermann, “Spectrally Efficient Compatible Single-SidebandModulation for OFDM Transmission With Direct Detection,” IEEE Photon.Technol. Letters 20, 670-672 (2008).

The ratio P_(LO)/P_(S) is of the order of 30 in reference [2].

This ratio reduces to values ranging from 4 to 8 in the case of theKramers-Kronig scheme.

Optical Digital Bandwidth Compensation P_(LO)/P_(s) IMDD R not possibleN/A Schmidt-reference [1] R possible  ~1 Schuster-reference [3] R/2 notpossible >>1 Randel-reference [2] R/2 possible >>1 Kramers-Kronig R/2possible  >1 (Current application)

Nonlinear Transmission Performance

In this section, we investigate the limitations imposed by the fibernonlinearity to the implementation of the Kramers-Kronig transceiver incoherent transmission systems of the kind considered in the previoussection. The main results of this investigation are presented in FIG. 5,which was obtained for a DWDM system with five 16 QAM channels at 24Gbaud. The channel spacing was set to 40 GHz. The BER of the channel ofinterest is plotted in FIG. 5 as a function of the total transmit power,which is the sum of the channel power and the local oscillator power.The upper panel 151 and the lower panel 152 refer to a one-span andtwo-span system, respectively, where in both cases a standard SMF wasassumed. Here too the various curves correspond to different values ofthe LO power, and CD was compensated after signal reconstruction.

As can be seen in the FIG. 5, the FEC threshold of 1.5×10⁻² is exceededfor a broad range of power levels. As expected, the BER improves withthe launched power, until it reaches an optimal value, after which itincreases as a result of growing nonlinear distortions. The effect ofthe local oscillator power is dual. On the one hand, it improves thecompliance with the Kramers-Kronig condition, which is beneficial forthe BER. But on the other hand, it enhances the nonlinear distortion,whose effect on the BER is adverse.

For this reason, the dependence of the peak BER on the LO power is notmonotonic. In the example of FIG. 5, the peak BER reduces by increasingthe LO power from 6 dB (P_(LO)=4P_(ch)) to 9 dB, and then it increaseswhen the LO power is raised beyond nine times the average power of thedata-carrying signal. In parallel, the range of channel powers for whichthe FEC requirement is satisfied shrinks fairly monotonically withincreasing LO power.

For comparison, we show in the same figure the BER of an 8 PAM systemoperated at the baudrate of 32 Gbaud, so as to provide the samethroughput. In this case the channel spacing was set to 50 GHz, andoptical CD compensation was implemented at the receiver. The fundamentalwaveform used was the same as in the case of 16 QAM modulation and eightequally spaced amplitude (not intensity) levels were used to encode theinformation. The figure shows that 8 PAM modulation (at least theidealized implementation that we simulated in this work) over-performs16 QAM in terms of optimum BER in the single-span configuration, but nosubstantial difference is visible in the two-span configuration. Weremind that the reduced performance of 16 QAM comes with a smallerspectral occupancy.

Referring again to FIG. 5—that illustrates the BER versus total transmitpower for the channel of interest of a DWDM system with five transmittedchannels. The filled markers were obtained for 16 QAM modulation basedon the use of the Kramers-Kronig scheme with various levels of the localoscillator power; the empty circles show the results obtained for 8 PAMmodulation. In the Kramers-Kronig scheme CD was compensated after signalreconstruction, whereas in the case of 8 PAM CD was compensatedoptically. Top and bottom panels differ by the number of spans.

FIG. 6 illustrates a Kramers-Kronig receiver that includes a photodiode21, analog to digital converter 30, an up-sampling module 41, alogarithm calculation module 42, a Hilbert transform processor 43 and adown-sampling module 44.

FIG. 6 also illustra es various signals such CW signal 11 and modulatedsignal 12, received signal Es 51, photocurrent I(t) 52, digitalrepresentation I(n/EB) 53 of the photocurrent (outputted by analog todigital converted 30), up-sampled digital signal 54 (outputted byup-sampling module 41), a Hilbert transformed signal Es(n/2 mb) 55(outputted by Hilbert transform processor 43) and a down-sampled Hilberttransformed signal Es(n/2 b) 56 (outputted by down-sampling module 44).

Extensions of the Kramers Kronig Receiver that Accommodate PolarizationMultiplexing

There may be provided at least one extension of theKramers-Kronig-receiver that are compatible with polarizationmultiplexed transmission.

In the case of the first extension, the scheme of FIG. 1A can becombined with polarization multiplexing provided that the receiver isequipped with a device that allows it to separate the two orthogonalpolarizations that were multiplexed at the transmitter. This operationcan be implemented by means of a controllable polarization rotatorfollowed by a Polarizing Beam Splitter (PBS). The controllable rotatorwill rotate the incoming signal's polarization until the average powersat the two ports of the PBS are equalized. When the powers at the twoPBS outputs are equal, the CW component is equally present in bothoutputs and each of the outputs of the PBS can be processed by aseparate (scalar) KK-receiver.

Since the bandwidth of the photocurrent I is twice larger than thebandwidth of the information carrying optical signal, the minimumsampling rate that is required according to Shannon Nyquist samplingtheorem is 2 B. The doubling of the bandwidth is consistent with thefact that the information that was previously encoded in acomplex-valued signal, is transferred by square-law detection into areal-valued signal without loss. The photocurrent samples are up-sampledand then the natural logarithm operation is performed. The up-samplingis required so as to accommodate the increase in bandwidth caused by thelogarithm operation. Subsequently, a Hilbert transform is applied so asto obtain the phase φ_(E) and the complex signal E_(S)(t), according toEquations (3) and (4). At this stage, the signal E_(S)(t) can bedown-sampled to the original sampling rate. The subsequent digitalprocessing of the received signal is identical to the one that is foundin standard coherent receivers, and is not detailed in the figure.

In the case of the second extension, a complex-valued polarizationmultiplexed signal 18 is transmitted through the fiber 210 along with acarrier (CW signal 202) aligned with the low-frequency edge of thedata-carrying signal (first modulated signal 201) in one of the twopolarizations (see FIG. 7A).

The detection in this scheme is performed with the help of a Stokesreceiver 220 which uses an optical hybrid. The advantages of theKramers-Kronig-Stokes receiver relative to standard coherent receptionsare: (1) It avoids the need for a local oscillator (2) It uses a simpleroptical hybrid (3) It requires only three (instead of four) analog todigital converters (ADCs).

The third implementation of the polarization multiplexed Kramers-Kronigreceiver that we present, requires a local oscillator, but the detectionitself is implemented with a single pair of photo-diodes (one perpolarization), completely avoiding the need for interferometric accuracyin the optical front-end (FIG. 7B). Correspondingly, it requires onlytwo ADCs.

The Kramers-Kronig-Stokes Polarization Multiplexed Receiver

In order to explain the principle of operation of theKramers-Kronig-Stokes receiver we start from describing thereconstruction of the complex envelopes of the two polarization channelsin a back-to-back configuration. Namely, assuming that the modulatedpolarization channels are x and y and that there is no rotation betweenthe transmitter and receiver (so that the x and y directions are easilyidentified by it). Subsequently we will explain what happens when thereceiver is far from the transmitter and the x and y directions of thetransmitter are not known to it.

In FIG. 7A the polarization multiplexed signal 18 includes secondmodulated signal 203 and first modulated signal 201. In FIG. 7A thesecond modulated signal 203 has a second polarization, the CW signal 202and the first modulated signal 201 have a first polarization (forexample—x-polarization) that differs from the second polarization (forexample y-polarization).

The transmitted fields are

E _(x)(t)=Ae ^(iπBt) +a _(x)(t)  (8)

E _(y)(t)=a _(y)(t)  (9)

Where a_(x)(t) and a_(y)(t) are the complex envelopes of the twoinformation carrying signals 201 and 203, whose spectral width is B. Theterm A represents the complex amplitude of a carrier tone transmitted atthe frequency −B/2 together with the x-polarized channel. The Stokesreceiver 220 reproduces the instantaneous Stokes vector of the field,which is given by the three Stokes receiver output signals S1 231, S2232 and S3 233.

S1=|Ex| ² −|Ey| ²  (10)

S2=2Re(Ex*Ey)  (11)

S3=−2Im(Ex*Ey).  (12)

The signal |Ex|² is extracted from the relation |Ex|²=0.5*(S0+S1), whereS0=|Ex|²+|Ey|² and the field Ex(t) is subsequently extracted using theKramers-Kronig reconstruction method described above. Once Ex(t) isknown, Ey(t) is extracted from the relation:

$\begin{matrix}{{{{\overset{\sim}{E}}_{y}(t)} = \frac{S_{2} - {iS}_{3}}{2{{\overset{\sim}{E}}_{x}^{*}(t)}}},} & (13)\end{matrix}$

Where the tilde is used to indicate that Ex and Ey denote thereconstructed fields. When the minimum-phase condition is fulfilled, thereconstruction is perfect and {tilde over (E)}x,y=Ex,y. Notice that theKramers-Kronig-procedure, through which {tilde over (E)}x(t) isreconstructed implies that its amplitude never crosses zero and hencethe division in Equation (13) can be performed with no problem, as wedemonstrate in the Section below. Note that when the data carryingsignals a_(x) and a_(y) have a zero mean and the same average power, thetime averaged Stokes vector is given by S_(t)=(|A|², 0, 0)^(T), where bythe superscript T we denote “transpose”. This property will be used inwhat follows.

When the receiver and the transmitter are not adjacent to each other,the signal undergoes random polarization rotations while propagating inthe fiber. In this case, the field extraction procedure can be carriedout as described earlier, provided that the S1 direction coincides withthe orientation of the time-average of the received Stokes vector.

We will denote this time averaged vector by {right arrow over (S)}_(TA).The procedure of field reconstruction is carried out as follows. First,one generates a rotation matrix M_(R) such that only the first componentof M_(R){right arrow over (S)}_(TA) is non-zero. Then, after thisrotation is applied to the received instantaneous Stokes vector {rightarrow over (S)}(t), E_(x) and E_(y) can be extracted as in the back toback case. It should be noted that the matrix M_(R) is not unique, asthe S2 and S3 directions can be arbitrarily chosen in the planeorthogonal to the S1 direction. This arbitrariness merely translatesinto a constant phase uncertainty in the recovered E_(y), which iseliminated by a standard phase recovery algorithm. An interestingconsequence of the Kramers-Kronig-Stokes procedure is that itintrinsically identifies the polarizations of the data carrying channelsand hence it does not require a MIMO algorithm for polarizationdemultiplexing.

In general, an ideal implementation of the Kramers-Kronig-Stokesreceiver (with perfectly square filters and with the minimum phasecondition satisfied) does not incur any noise penalty with respect to acoherent receiver.

We note that as long as the minimum-phase condition is satisfied, thedescribed procedure is not affected in any way by the presence ofchromatic, or polarization mode dispersion.

These phenomena merely imply that the reconstructed fields a_(x)(t) anda_(y)(t) are distorted versions of the waveforms that were produced bythe transmitter. These distortions are not different from those existingin coherent transmission schemes and they are to be treated accordingly.

S1, S2 and S3 are fed to three ADCs 251, 242 and 243 respectively thatgenerate digital representations of S1, S2 and S3 that are fed todigital processor 244. The ADCs and the digital processor arecollectively denoted ADC DSP 240. ADC DSP 240 may apply theKramers-Kronig algorithm, as well as the functionalities of a coherentreceiver's digital processor.

ADC DSP 240 may perform the following:

-   -   a. Find the time averages values of S1, S2, and S3 which        together form a time averaged Stokes vector {right arrow over        (S)}_(TA).    -   b. Find a rotation matrix M_(R) (which is real valued symmetric        matrix with orthogonal columns) such that M_(R){right arrow over        (S)}_(TA) is a vector whose second and third components are        zeros).    -   c. Apply the matrix M_(R) to the Stokes vector {right arrow over        (S)}(t) (whose components are S₁(t) S₂(t) and S₃(t)) such that a        rotated Stokes vector {right arrow over (S)}_(R)(t)=M_(R){right        arrow over (S)}(t) is formed. The three components of {right        arrow over (S)}_(R)(t) are denoted by S_(R,1)(t), S_(R,2)(t),        and S_(R,3)(t).    -   d. Define S₀=|{right arrow over (S)}_(R)|=√{square root over        (S_(R,1) ²+S_(R,2) ²+S_(R,3) ²)}.    -   e. Define I(t)=½(S₀+S_(R,1))    -   f. Apply a Kramers-Kronig algorithm (in one of its above listed        versions) to I(t) in order to extract E_(x)(t). This relies on        the steps        -   i. Define ϕ_(rec)(t)=½Hilbert{log[I(t)]}, where the digital            implementation of this stage may imply the need for            up-sampling.        -   ii. Extract E_(x)(t) from E_(x)(t)=√{square root over            (I(t))}e^(iϕ) ^(rec) ^((t)), where A may be estimated as the            time average of √{square root over (I(t))}e^(iϕ) ^(rec)            ^((t)).        -   iii. Extract E_(y)(t) from the relation E_(y)(t)            =(S_(R,2)−iS_(R,3))/E_(x).    -   g. Perform additional operations such as phase recovery,        chromatic dispersion compensation, and possibly nonlinear        distortion compensation.

Kramers-Kronig Polarization Multiplexing with Two Photo-Diodes

In this scheme, a local oscillator is used by the receiver. The use of alocal oscillator does not necessarily require a dedicated laser, as thesignal generated by the transmitter's laser may be split in order toaccommodate both functionalities.

As illustrated in FIG. 7B, the received signal and the local oscillatorare mixed so that the signals that are photo-detected along the x and ypolarization components are

E _(x)(t)=Ae ^(iπBt) +a _(x)(t)  (7)

E _(y)(t)=Ae ^(iπBt) +a _(y)(t)  (8)

Where in this case a_(x)(t) and a_(y)(t) represent the x and ypolarization components of the incoming optical signal. Provided that Ais large enough in order to fulfill the minimum phase condition, thesignals E_(x)(t) and E_(y)(t) are both minimum phase, in which casea_(x)(t) and a_(y)(t) can be extracted from the respective intensities|E_(x)|² and |E_(y)|². The extraction of the transmitted data froma_(x)(t) and a_(y)(t) relies on the standard digital processing used ingeneric coherent transmission.

FIG. 7B illustrates a polarization multiplexed signal 17 that includessecond modulated signal 203 and first modulated signal 201. In FIG. 7Bthe second modulated signal 203 has a second polarization, the firstmodulated signal 201 has a first polarization (forexample—x-polarization) that differs from the second polarization (forexample y-polarization).

The polarization multiplexed signal 17 is fed to a polarizationdemultiplexing circuit 270 (includes polarizing beam splitter (PBS) 212)that is configured to receive polarization-multiplexed signal 17 and tooutput a first polarization component of the polarization-multiplexedsignal to first coupler 281 and a second polarization component of thepolarization-multiplexed signal to second coupler 282.

The first coupler 281 and the second coupler 282 also receive a CWsignal from polarizing beam splitter PBS 216. PBS 216 is fed by localoscillator 214.

Each one of the first and second couplers adds the CW signal to thepolarization components they receive and provides a first intermediatesignal to the first photodiode 251 and a second intermediate signal tothe second photodiode, respectively.

First photodiode 251 is configured to receive the first intermediatesignal and output a first photocurrent that represents the firstintermediate signal.

Second photodiode 252 is configured to receive the second intermediatesignal and output a second photocurrent that represents the secondintermediate signal.

There may or may not be a frequency gap between the continuous wave andeach one of the first and second intermediate signals. The frequency gapis to facilitate signal reconstruction and may be smaller than thebandwidth of the first modulated signal and the bandwidth of the secondmodulated signal. The frequency gap may be any fraction of the bandwidthof any one of the first and second modulated signals—for example smallerthan one half of the bandwidth of any one of the first and secondmodulated signals.

First and second photodiodes may be followed by ADC DSP 260.

ADC DSP 260 may include (i) first ADC 261 that is configured to generatea digital representation of the first photocurrent; (ii) second ADC 262that is configured to generate a digital representation of the secondphotocurrent; (ii) a digital processor 264 that is configured to: (i)process the digital representation of the first photocurrent to providea reconstructed phase of the first input signal and a reconstructedcomplex amplitude of the first input signal, wherein the processing isbased on the Kramers-Kronig relationship related to the firstpolarization component; and (ii) process the digital representation ofthe second photocurrent to provide a reconstructed phase of the secondinput signal and a reconstructed amplitude of the second input signal,wherein the processing is based on the Kramers-Kronig relationshiprelated to the second polarization component.

For each one of the two paths, the digital processor may perform thefollowing operations:

-   -   i. Up-sample the digital representation of the photocurrent        I_(n)(t)    -   ii. Calculate a reconstructed phase        ϕ_(E,n)(t)=0.5*Hilbert{log(I_(n)(t))}.    -   iii. Calculate a reconstructed modulated signal        E_(s,n)(t)=√{square root over (I(t))}e^(iϕ) ^(E,n)        ^((t))−E_(0,n), where E_(0,n) is the complex envelope of the CW        signal, which may be evaluated as the time average √{square root        over (I(t))}e^(iϕ) ^(E,n) ^((t))    -   iv. Perform coherent processing steps.    -   In steps i-iv the subscript n is either 1 or 2 (or x or y),        depending on the path to which it refers.

Considerations concerning the requirements on the local oscillator'spower as well as the effect of fiber dispersion and nonlinearities arepresented in the Section below.

A hidden assumption that was made for the simplicity of illustration isthat the LOs in the x and y polarizations have the same phase (itscomplex amplitude is denoted by E₀ in step iii above). In practice, thephases may differ, as the optical paths of the two polarizationcomponents of the LO and the data carrying signal are not identicalafter polarization splitting.

This is not an issue that impedes the implementation of the scheme, asthis phase difference is standardly eliminated digitally in a MIMOdemultiplexing algorithm, of the kind used in all coherent receivers.

As in the Kramers-Kronig-Stokes scheme, the operating principle of thereceiver path of FIG. 7B (also referred to as the two photodiode (2 PD)approach) is not undermined by the presence of noise or propagationdistortions.

Numerical Validation

In order to validate the proposed schemes, we simulated a 100 km linkimplemented over standard single-mode fiber (dispersion coefficient D=17ps/nm/km, and nonlinearity coefficient γ=1.3 W⁻¹km⁻¹, loss of 0.2 dB perkm, and negligible polarization mode dispersion and polarizationdependent loss). The system is assumed to operate at 32 Gbaud with16-QAM modulation and raised cosine pulses with a roll-off factor of0.05. We assume 11 WDM channels separated by 50 GHz and evaluate theperformance of the central channel. Each simulation was performed with2¹⁵ symbols and 50 such simulations are performed for every displayedBER point. In all cases, the BER is evaluated under the assumption ofGrey coding. In both of the examined schemes we use a 12-th ordersuper-Gaussian optical filter whose 3-dB bandwidth was 48 GHz.

We set the central frequency of the filter to be 5.3 GHz higher than thecenter frequency of the channels (i.e. it is higher by 22.1 GHz than thefrequency of the carrier in the x polarization). All simulations rely onthe split-step solution of the Manakov equation and are performedwithout polarization mode dispersion. A random rotation of thepolarization state prior to reception is applied in all cases.

A. Validating the Kramers-Kronig-Stokes Scheme

FIG. 8 shows the results obtained with the Kramers-Kronig-Stokes scheme.

The upper panel 161 of FIG. 8 illustrates the BER as a function of theequivalent OSNR, in the regime linear propagation. The filled and emptymarkers refer to the x- and y-polarized channels, respectively. Theresults are shown in the cases where the power |A²| of the CW tone is 3Ps, 4 Ps, and 5 Ps, where P_(S) is the total power of the data-carryingsignal (in both polarizations). The dashed curve corresponds to the caseof an ideal coherent receiver. (b) The BER versus the total transmittedpower in a 100 km SMF link. See main text for details. Suddeninterruption of a BER curve is due to absence of errors in the recoveryof the simulated 215 symbol sequence.

The upper panel 161 of FIG. 8 ignores the effects of nonlinearity bysetting γ to 0, and show the BER as a function of the equivalent OSNR,which is the OSNR that would characterize an equivalent system using acoherent receiver. Namely, the equivalent OSNR is the total signal power(without the CW tone) divided by the noise power in a bandwidth of 0.1nm. The results are shown for the cases where the carrier power |A|² isequal to 3, 4, and 5 times the total power of the signal. The filled andthe empty markers correspond to the x and y polarization channels,respectively, and the dashed line shows the result corresponding to anideal coherent receiver. As is obvious in the figure, at sufficientlyhigh OSNR, the BER saturates. Two observations can be readily made.

One is that the BER of the x-polarized channel always saturates earlierthan that of the y-polarized channel. The second is that the stronger|A|², the higher is the SNR at which the saturation occurs. Theexplanation for the latter phenomenon is that errors can be caused bynoise, or by incorrect reconstruction as a result of the fact that theminimum-phase condition may be violated. The errors observed in the lowOSNR case are caused by noise, but when the OSNR increases to the pointwhere noise-induced errors become rarer than the errors caused byincorrect reconstruction, the BER saturates. As the probability ofreconstruction errors reduces with the intensity of the carrier, thesaturation occurs at higher OSNR values when |A|² is increased.

The reason for the fact that the y polarized channel saturates laterthan its x-polarized counterpart is a little subtler and can beexplained as follows.

Denoting the reconstruction error of the x-polarized field by e(t), sothat Ë_(x)(t)=E_(x)(t)+ε(t), Equation (13) can be expressed as:

$\begin{matrix}{{\overset{\sim}{E}}_{y} = {{\frac{1}{2}\frac{S_{2} - {iS}_{3}}{E_{x}^{*} + \epsilon^{*}}} \simeq {E_{y} - {\frac{E_{y}}{E_{x}^{*}}{\epsilon^{*}.}}}}} & (14)\end{matrix}$

Implying that the variance of the reconstruction error of Ey(t) issmaller than that of Ex by a factor of 1+2|A|²/Ps with Ps=E[|ax|²+|ay|²]denoting the average power of the data carrying signal. Note that in allcases, the saturation occurs at BER levels well below the relevantthreshold level of 10⁻² and hence the scheme can be safely operated witheven with |A|²=3 Ps, implying that the total launched power with theKramers-Kronig-Stokes scheme is 6 dB higher than in the coherent case.

In order to examine the effect of nonlinearity, we show in the lowerpanel 162 of FIG. 8 the BER as a function of the launched power per WDMchannel (in both polarizations, i.e. Ps+|A|²). Once again, the ypolarization performs better than x, which in this case results bothfrom the reduced reconstruction error and from the fact that thenonlinear effect of the carrier is stronger in the case of thex-polarized channel, which is parallel to it. Nonetheless, when |A|²=3Ps the saturation in the case of the x polarization is dominated byreconstruction errors, as can be seen from the fact that the BER isalmost identical to its value in the linear case, and is safely lowerthan typical FEC thresholds. This suggests once again, that the setting|A|²=3 Ps is adequate for a well operating Kramers-Kronig-Stokesreceiver.

B. Validating the Two Photo-Diode Scheme

FIGS. 9A and 9B show (panels 171 and 172) the BER in the two photo-diodeconfigurations. FIGS. 9A and 9B correspond to the case of linear andnonlinear transmission. Owing to the symmetry of the scheme, the twopolarization channels perform identically, and their performance isidentical to that of the x-polarized channel in theKramers-Kronig-Stokes implementation. In order for the comparison in thenonlinear case to be meaningful, we plot in FIG. 10 (panel 181) the BERas a function of the equivalent OSNR (which is identical to the actualOSNR in the 2 PD case) for the x-polarized channels in the two receiverschemes. This time the empty markers correspond to theKramers-Kronig-Stokes receiver, whereas the filled markers correspond tothe Kramers-Kronig-2 PD scheme. As is evident from the figure, theKramers-Kronig-2 PD scheme is more tolerant to the fiber nonlinearity,owing to the smaller total launch power. Moreover, as the power of theCW signal is increased (approximately above the value of 10 Ps), theperformance of the Kramers-Kronig-2 PD scheme approaches that ofcoherent homodyne, since the increased CW signal power improves thesignal reconstruction quality without affecting the magnitude of thepropagation-induced nonlinear distortion

Discussion

Both presented approaches for extending the Kramers-Kronig-scheme toaccommodate polarization multiplexed transmission were shown to producepromising results, with error-rates well below the frequently quoted0.01 FEC threshold. For adequate performance, the power that needs to beallocated for the carrier (the local oscillator) is of the order of 3 Psand 6 Ps in the Kramers-Kronig-Stokes and the 2 PD scheme, respectively.The disadvantage of requiring a LO signal in the latter case is notsignificant, given that it can be extracted from the transmission laser.On the other hand, the fact that the 2 PD scheme involves only two ADCsand does not require an optical hybrid implies a notable reduction incost.

As was pointed out above, the Kramers-Kronig-Stokes procedure does notrequire a MIMO algorithm for polarization demultiplexing. On the otherhand, the existence of different performance for the two polarizationchannels may be considered a disadvantage.

A simple modification that remedies this is one where the carrier'sstate of polarization is set to be at 45 degrees between the x and ypolarizations. In this case, half of the carrier power accompanies eachone of the two polarization channels, and the Kramers-Kronig-receiverreconstructs the field components polarized at _45 degrees. With thismodification, the performance of the two polarization channels isequalized, but the implementation of a MIMO algorithm for polarizationdemultiplexing can no longer be avoided (as in the coherent orKramers-Kronig-2 PD cases).

The configuration illustrated in FIG. 7A and 7B may be regarded as anextension to the configuration of FIGS. 1A and 1B, the extensionsaccommodate for polarization multiplexed transmission. The first schemecombines a Stokes receiver with Kramers-Kronig-processing, and requiresthe transmission of a CW tone together with one of the polarizationchannels. The second scheme uses only two photo-diodes for detection,one per polarization, and does not require the transmission of a CWtone. On the other hand, it relies on the availability of a localoscillator. We demonstrated the performance of the two schemes fortransmission settings inspired by the needs of inter-data centercommunications.

FIG. 11 illustrates a transceiver that includes a reception path 720 anda transmission path 770.

The reception path 720 includes a receiver (Rx) path laser 721, acombiner 781 for adding the CW signal from the Rx path laser 721 to areceived signal (such as modulated signal 701) to provide intermediatesignal 710, a photodiode 751 that outputs a photocurrent that representsthe modulated signal 701, an ADC 761 that performs an analog to digitalconversion of the photocurrent and a digital processor 764 that mayprocess the digital signal from the ADC based on the Kramers-Kronigrelationship related to the modulated signal. The reception path 720 mayinclude any of the components illustrated, for example, in FIG. 1B.

The transmission path 770 includes a transmission (Tx) path laser 780that is followed by a complex modulator 772 that outputs transmittedsignal 703.

The received signal 701 has a bandwidth B and a central frequency F0.The intermediate signal includes a CW signal (provided by Rx path laser721) that has a frequency that does not exceed F0-B/2—in order to bepositioned to the left of the received signal 701.

The transmitted signal 703 a bandwidth B and a central frequency F0.Accordingly—the CW signal provided by Tx path laser 780 has a frequencyof F0.

The different frequencies of the CW signals generated by Rx path laser721 and Tx path laser 780 requires using two separate lasers.

FIG. 12 illustrates a transceiver that includes a reception path 820 anda transmission path 880.

This transceiver includes a single laser—TX/RX laser 881.

TX/RX laser 881 generates a CW signal (at frequency F0) that is fed to(a) the complex modulator 882 of the transmission path 880, and to thereception path 820.

The reception path 820 include a combiner 881 that combines the CWsignal from the TX/RX laser 881 and a modulated signal 801 (from acommunication line) to provide a combined signal that is fed to a firstoptical filter 891 and a second optical filter 892 of the receptionpath.

The modulated signal 801 includes two sidebands 8011 and 8012 with aguard band 805 in the middle between ω=F0−δ and ω=F0+δ. The value of δis anything between 0 and a certain fraction of B (for example B/4), andit may be chosen to have the lowest value that still accommodatesseparation between the two sidebands 8011 and 8012 by using firstoptical filter 891 and a second optical filter 892 or any other means ofoptical filtering.

The first optical filter 891 outputs a first intermediate signal 811that includes the CW signal from TX/RX laser 881 and sideband 8011.

The second optical filter 892 outputs a second intermediate signal 812that includes the CW signal from TX/RX laser 881 and sideband 8012.

First photodiode 851 outputs a first photocurrent that represents firstintermediate signal 811, first ADC 861 performs an analog to digitalconversion of the first photocurrent and first digital processor 864 mayprocess the digital signal from the first ADC based on theKramers-Kronig relationship related to the modulated signal.

Second photodiode 852 outputs a second photocurrent that representssecond intermediate signal 812, second ADC 862 performs an analog todigital conversion of the second photocurrent and second digitalprocessor 865 may process the digital signal from the second ADC basedon the Kramers-Kronig relationship related to the modulated signal.

The reception path 820 may include any of the components illustrated,for example, in FIG. 1B.

The bandwidth B can be doubled relative to the implementation in FIG.11, provided that the corresponding bandwidth can be supported by thetransmitter.

The CW signal may be added after the optical filtering, that is, at theoutput of each one of first optical filter 891 and second optical filter892. This configuration requires one splitter for the RX/TX laser andtwo couplers at the output of the first and second optical filters. Thisconfiguration allows more flexibility on the design of the opticalfilters.

FIG. 13 illustrates a transceiver according to an embodiment of theinvention.

The transceiver of FIG. 13 includes reception path 910 and atransmission path 970.

The transmission path 970 may include TX/RX laser 971 and complexmodulator 972.

The reception path 910 includes PBS 901, PBS 902, splitters 911, 912,913 and 914, combiners 921, 922, 923 and 924, photodiodes 941, 942, 943and 944, ADCs 951, 952, 953 and 954 and digital processor 960.

A polarization multiplexed signal 890 is received by PBS 901.

The polarization multiplexed signal 890 includes a combination of (a) afirst polarized modulated signal that includes two sidebands and with aguard band in the middle between ω=F0−δ and ω=F0+δ (such as modulatedsignal 801 of FIG. 12), and of (b) a second polarized modulated signalthat includes two sidebands and with a guard band in the middle betweenω=F0−δ and ω=F0+δ (such as modulated signal 801 of FIG. 12).

A first polarization component of the polarization multiplexed signal890 is sent by PBS 901 to splitter 911 and then to first and secondoptical filters OF1 931 and OF2 932.

First optical signal OF1 931 outputs a first sideband of the firstpolarization component of the polarization multiplexed signal 890.

Second optical signal OF2 932 outputs a second sideband of the firstpolarization component of the polarization multiplexed signal 890.

Combiner 921 combines a first polarization CW signal with the firstsideband of the first polarization component of the polarizationmultiplexed signal 890 to provide a first intermediate signal that has afield E_(x,1) 1891 and is fed to first photodiode 941.

Combiner 922 combines a first polarization CW signal with the secondsideband of the first polarization component of the polarizationmultiplexed signal 890 to provide a second intermediate signal that hasa field E_(x,2) 2892 and is fed to second photodiode 942.

First photodiode 941 outputs a first photocurrent I_(x,1)(t) 991 that isanalog to digital converted by ADC 951 and is fed to digital processor960.

Second photodiode 942 outputs a second photocurrent I_(x,2)(t) 992 thatis analog to digital converted by ADC 952 and is fed to digitalprocessor 960.

Combiner 923 combines a second polarization CW signal with the firstsideband of the second polarization component of the polarizationmultiplexed signal 890 to provide a third intermediate signal that has afield E_(x,3) 893 and is fed to third photodiode 943.

Combiner 924 combines the second polarization CW signal with the secondsideband of the second polarization component of the polarizationmultiplexed signal 890 to provide a fourth intermediate signal that hasa field E_(x,4) 894 and is fed to fourth photodiode 944.

Third photodiode 943 outputs a third photocurrent I_(x,3)(t) 993 that isanalog to digital converted by ADC 953 and is fed to digital processor960.

Fourth photodiode 944 outputs a fourth photocurrent LAO 994 that isanalog to digital converted by ADC 954 and is fed to digital processor960.

Digital processor 960 may process the digital signals from ADCs 951-954using one or more Kramers-Kronig relationships.

The processing may include

-   -   a. Extracting E_(x,1) from I_(x,1) using a Kramers-Kronig        algorithm.    -   b. Extracting E_(x,2) from I_(x,2) using the Kramers-Kronig        algorithm.    -   c. Extracting E_(y,1) from I_(y,1) using the Kramers-Kronig        algorithm.    -   d. Extracting E_(y,2) from I_(y,2) using the Kramers-Kronig        algorithm.    -   e. Applying any coherent processing to the vectors

$\begin{bmatrix}E_{x,1} \\E_{y,1}\end{bmatrix}\mspace{14mu} {{and}\mspace{14mu}\begin{bmatrix}E_{x,2} \\E_{y,2}\end{bmatrix}}$

FIG. 14 illustrates method 1400 according to an embodiment of theinvention.

Various steps of method 1400 may be executed by each one of theKramers-Kronig receivers and/or the Kramers-Kronig transceiversillustrated in any of the figures and/or in the specification.

Method 1400 may include step 1410 of receiving a signal.

Step 1410 may be followed by step 1420 of reconstructing the signalbased on at least one Kramers-Kronig relationship related to the signal.

Method 1400 may also include generating a transmitted signal. Thegeneration of the transmitted signal may be related to variousKramers-Kronig transceivers such as those illustrated in FIGS. 11-13.

Method 1400 may include, for example, any combination of the steps ofparagraphs [0031]-[0059].

The invention may also be implemented in a computer program for runningon a computer system, at least including code portions for performingsteps of a method according to the invention when run on a programmableapparatus, such as a computer system or enabling a programmableapparatus to perform functions of a device or system according to theinvention. The computer program may cause the storage system to allocatedisk drives to disk drive groups.

A computer program is a list of instructions such as a particularapplication program and/or an operating system. The computer program mayfor instance include one or more of: a subroutine, a function, aprocedure, an object method, an object implementation, an executableapplication, an applet, a servlet, a source code, an object code, ashared library/dynamic load library and/or other sequence ofinstructions designed for execution on a computer system. Forexample—the computer program may include a list of instructions thatcauses any of the mentioned above digital processors to perform any ofthe mentioned above processing.

The computer program may be stored internally on a non-transitorycomputer readable medium. All or some of the computer program may beprovided on computer readable media permanently, removably or remotelycoupled to an information processing system. The computer readable mediamay include, for example and without limitation, any number of thefollowing: magnetic storage media including disk and tape storage media;optical storage media such as compact disk media (e.g., CD-ROM, CD-R,etc.) and digital video disk storage media; nonvolatile memory storagemedia including semiconductor-based memory units such as flash memory,EEPROM, EPROM, ROM; ferromagnetic digital memories; MRAM; volatilestorage media including registers, buffers or caches, main memory, RAM,etc.

A computer process typically includes an executing (running) program orportion of a program, current program values and state information, andthe resources used by the operating system to manage the execution ofthe process. An operating system (OS) is the software that manages thesharing of the resources of a computer and provides programmers with aninterface used to access those resources. An operating system processessystem data and user input, and responds by allocating and managingtasks and internal system resources as a service to users and programsof the system.

The computer system may for instance include at least one processingunit, associated memory and a number of input/output (I/O) devices. Whenexecuting the computer program, the computer system processesinformation according to the computer program and produces resultantoutput information via I/O devices.

In the foregoing specification, the invention has been described withreference to specific examples of embodiments of the invention. It will,however, be evident that various modifications and changes may be madetherein without departing from the broader spirit and scope of theinvention as set forth in the appended claims.

Moreover, the terms “front,” “back,” “top,” “bottom,” “over,” “under”and the like in the description and in the claims, if any, are used fordescriptive purposes and not necessarily for describing permanentrelative positions. It is understood that the terms so used areinterchangeable under appropriate circumstances such that theembodiments of the invention described herein are, for example, capableof operation in other orientations than those illustrated or otherwisedescribed herein.

The connections as discussed herein may be any type of connectionsuitable to transfer signals from or to the respective nodes, units, ordevices, for example via intermediate devices. Accordingly, unlessimplied or stated otherwise, the connections may for example be directconnections or indirect connections. The connections may be illustratedor described about being a single connection, a plurality ofconnections, unidirectional connections, or bidirectional connections.However, different embodiments may vary the implementation of theconnections. For example, separate unidirectional connections may beused rather than bidirectional connections and vice versa. Also,plurality of connections may be replaced with a single connection thattransfers multiple signals serially or in a time multiplexed manner.Likewise, single connections carrying multiple signals may be separatedout into various connections carrying subsets of these signals.Therefore, many options exist for transferring signals.

Although specific conductivity types or polarity of potentials have beendescribed in the examples, it will be appreciated that conductivitytypes and polarities of potentials may be reversed.

Each signal described herein may be designed as positive or negativelogic. In the case of a negative logic signal, the signal is active lowwhere the logically true state corresponds to a logic level zero. In thecase of a positive logic signal, the signal is active high where thelogically true state corresponds to a logic level one. Note that any ofthe signals described herein may be designed as either negative orpositive logic signals. Therefore, in alternate embodiments, thosesignals described as positive logic signals may be implemented asnegative logic signals, and those signals described as negative logicsignals may be implemented as positive logic signals.

Furthermore, the terms “assert” or “set” and “negate” (or “deassert” or“clear”) are used herein when referring to the rendering of a signal,status bit, or similar apparatus into its logically true or logicallyfalse state, respectively. If the logically true state is a logic levelone, the logically false state is a logic level zero. And if thelogically true state is a logic level zero, the logically false state isa logic level one.

Those skilled in the art will recognize that the boundaries betweenlogic blocks are merely illustrative and that alternative embodimentsmay merge logic blocks or circuit elements or impose an alternatedecomposition of functionality upon various logic blocks or circuitelements. Thus, it is to be understood that the architectures depictedherein are merely exemplary, and that in fact many other architecturesmay be implemented which achieve the same functionality.

Any arrangement of components to achieve the same functionality iseffectively “associated” such that the desired functionality isachieved. Hence, any two components herein combined to achieve aparticular functionality may be seen as “associated with” each othersuch that the desired functionality is achieved, irrespective ofarchitectures or intermedial components. Likewise, any two components soassociated can also be viewed as being “operably connected,” or“operably coupled,” to each other to achieve the desired functionality.

Furthermore, those skilled in the art will recognize that boundariesbetween the above described operations merely illustrative. The multipleoperations may be combined into a single operation, a single operationmay be distributed in additional operations and operations may beexecuted at least partially overlapping in time. Moreover, alternativeembodiments may include multiple instances of a particular operation,and the order of operations may be altered in various other embodiments.

Also for example, in one embodiment, the illustrated examples may beimplemented as circuitry located on a single integrated circuit orwithin a same device. Alternatively, the examples may be implemented asany number of separate integrated circuits or separate devicesinterconnected with each other in a suitable manner.

Also for example, the examples, or portions thereof, may implemented assoft or code representations of physical circuitry or of logicalrepresentations convertible into physical circuitry, such as in ahardware description language of any appropriate type.

Also, the invention is not limited to physical devices or unitsimplemented in non-programmable hardware but can also be applied inprogrammable devices or units able to perform the desired devicefunctions by operating in accordance with suitable program code, such asmainframes, minicomputers, servers, workstations, personal computers,notepads, personal digital assistants, electronic games, automotive andother embedded systems, cell phones and various other wireless devices,commonly denoted in this application as ‘computer systems’.

However, other modifications, variations and alternatives are alsopossible. The specifications and drawings are, accordingly, to beregarded in an illustrative rather than in a restrictive sense.

In the claims, any reference signs placed between parentheses shall notbe construed as limiting the claim. The word ‘comprising’ does notexclude the presence of other elements or steps then those listed in aclaim. Furthermore, the terms “a” or “an,” as used herein, are definedas one or more than one. Also, the use of introductory phrases such as“at least one” and “one or more” in the claims should not be construedto imply that the introduction of another claim element by theindefinite articles “a” or “an” limits any particular claim containingsuch introduced claim element to inventions containing only one suchelement, even when the same claim includes the introductory phrases “oneor more” or “at least one” and indefinite articles such as “a” or “an.”The same holds true for the use of definite articles. Unless statedotherwise, terms such as “first” and “second” are used to arbitrarilydistinguish between the elements such terms describe. Thus, these termsare not necessarily intended to indicate temporal or otherprioritization of such elements. The mere fact that certain measures arerecited in mutually different claims does not indicate that acombination of these measures cannot be used to advantage.

While certain features of the invention have been illustrated anddescribed herein, many modifications, substitutions, changes, andequivalents will now occur to those of ordinary skill in the art. It is,therefore, to be understood that the appended claims are intended tocover all such modifications and changes as fall within the true spiritof the invention.

1. A Kramers-Kronig receiver, comprising a reception path; wherein thereception path comprises: a Stokes receiver that is configured toreceive a polarization-multiplexed signal and to output a Stokes vector;wherein the polarization-multiplexed signal comprises a first modulatedsignal, a second modulated signal and a continuous wave signal; whereinthe first modulated signal is of a first polarization; wherein thesecond modulated signal is of a second polarization; wherein thecontinuous wave signal is of the first modulation or of the secondmodulation; a set of analog to digital converters that are configured togenerate a digital representation of the Stokes vector; and a digitalprocessor that is configured to process the digital representation ofthe Stokes vector to provide a reconstructed polarization-multiplexedsignal, wherein the processing is based on a Kramers-Kronig relationshiprelated to the polarization-multiplexed signal.
 2. The Kramers-Kronigreceiver according to claim 21 wherein the polarization-multiplexedsignal has a first polarization field and a second polarization field;wherein the Stokes vector comprises a first Stokes receiver outputsignal that is indicative of (a) a square of an absolute value of thefirst polarization field minus (h) a square of an absolute value of thesecond polarization field.
 3. The Kramers-Kronig receiver according toclaim 21 wherein the Stokes vector comprises a second Stokes receiveroutput signal that is indicative of a real part of a given product of amultiplication of the first polarization field by the secondpolarization field, and a third Stokes receiver output signal that isindicative of an imaginary part of the given product.
 4. TheKramers-Kronig receiver according to claim 23 wherein the digitalprocessor is configured to: a. calculate a rotation matrix that onceapplied to a time averaged Stokes vector provides a rotated vector thathas a first non-zero element, a zero valued second element and a zerovalued third element; b. apply the rotation matrix on the Stokes vectorto provide a rotated Stokes vector; and; c. calculate the reconstructedpolarization-multiplexed signal based on the rotated Stokes vector.
 5. AKramers-Kronig transceiver, comprising a reception path and atransmission path and a local oscillator that is configured to generatea continuous wave (CW) signal; wherein the transmission path comprises acomplex modulator that is configured to receive the CW signal and isconfigured to modulate the CW signal to generate a transmitted signalthat is transmitted to a communication link; wherein the reception pathcomprises: a combiner that is configured to add the CW signal with areceived signal from the communication link to provide a combinedsignal; wherein the received signal comprises a first sideband and asecond sideband that are spaced apart from each other by a guard band;wherein the CW signal has a frequency that is included in the guardhand; a first optical filter that is configured to filter the combinedsignal and output a first intermediate signal that comprises the CWsignal and the first sideband; a second optical filter that isconfigured to filter the combined signal and output a secondintermediate signal that comprises the CW signal and the secondsideband; a first photodiode that is configured to receive the firstintermediate signal and output a first photocurrent that represents thefirst intermediate signal; a second photodiode that is configured toreceive the second intermediate signal and output a second photocurrentthat represents the second intermediate signal; a first analog todigital converter that is configured to generate a digitalrepresentation of the first photocurrent; a second analog to digitalconverter that is configured to generate a digital representation of thesecond photocurrent; and a digital processor that is configured toprocess the digital representation of the first photocurrent and thedigital representation of the second photocurrent to provide areconstructed received signal, wherein the processing is based on aKramers-Kronig relationship related to the received signal.
 6. TheKramers-Kronig receiver according to claim 25, wherein the guard banddoes not exceed a portion of a bandwidth of the first sideband.
 7. AKramers-Kronig transceiver, comprising a reception path and atransmission path and a local oscillator that is configured to generatea continuous wave (CW) signal; wherein the transmission path comprises acomplex modulator that is configured to receive the CW signal and isconfigured to modulate the CW signal to generate a transmitted signalthat is transmitted to a communication link; wherein the reception pathcomprises: a polarization demultiplexing circuit that is configured toreceive a polarization-multiplexed signal and to output a firstpolarization component of the polarization-multiplexed signal and asecond polarization component of the polarization-multiplexed signal; afirst optical filter that is configured to filter the first polarizationcomponent and output a first sideband of the first polarizationcomponent; a second optical filter that is configured to filter thefirst polarization component and output a second sideband of the firstpolarization component; wherein the first and second sidebands of thefirst polarization component are spaced apart from each other by a guardband; a first combiner that is configured to add the CW signal with thefirst sideband of the first polarization component to provide a firstcombined signal; a second combiner that is configured to add the CWsignal with the second sideband of the first polarization component toprovide a second combined signal; a first photodiode that is configuredto receive the first combined signal and output a first photocurrentthat represents the first combined signal; a second photodiode that isconfigured to receive the second combined signal and output a secondphotocurrent that represents the second combined signal; a first analogto digital converter that is configured to generate a digitalrepresentation of the first photocurrent; a second analog to digitalconverter that is configured to generate a digital representation of thesecond photocurrent; a third optical filter that is configured to filterthe second polarization component and output a first sideband of thesecond polarization component; a fourth optical filter that isconfigured to filter the second polarization component and output asecond sideband of the second polarization component; wherein the firstand second sidebands of the second polarization component are spacedapart from each other by the guard band; a third combiner that isconfigured to add the CW signal with the first sideband of the secondpolarization component to provide a third combined signal; a fourthcombiner that is configured to add the CW signal with the secondsideband of the second polarization component to provide a fourthcombined signal; a third photodiode that is configured to receive thethird combined signal and output a third photocurrent that representsthe third combined signal; a fourth photodiode that is configured toreceive the fourth combined signal and output a fourth photocurrent thatrepresents the fourth combined signal; a third analog to digitalconverter that is configured to generate a digital representation of thethird photocurrent; a fourth analog to digital converter that isconfigured to generate a digital representation of the fourthphotocurrent; and a digital processor that is configured to process thedigital representation of the first photocurrent, the secondphotocurrent, the third photocurrent and the fourth photocurrent toprovide a reconstructed polarization-multiplexed signal, wherein theprocessing is based on a Kramers-Kronig relationship related to thepolarization-multiplexed signal.
 8. A method, comprising: receiving, bya Stokes receiver of a reception path of Kramers-Kronig receiver, apolarization-multiplexed signal; outputting by the Stokes receiver, aStokes vector; wherein the polarization-multiplexed signal comprises afirst modulated signal, a second modulated signal and a continuous wavesignal; wherein the first modulated signal is of a first polarization;wherein the second modulated signal is of a second polarization; whereinthe continuous wave signal is of the first modulation or of the secondmodulation; performing an analog to digital conversion, by a set ofanalog to digital converters, of the Stokes vector to provide a digitalrepresentation of the Stokes vector; and processing, by a digitalprocessor of the reception path of Kramers-Kronig receiver, the digitalrepresentation of the Stokes vector to provide a reconstructedpolarization-multiplexed signal, wherein the processing is based on aKramers-Kronig relationship related to the polarization-multiplexedsignal.
 9. A method, comprising: generating by a local oscillator acontinuous wave (CW) signal; modulating, by a complex modulator, the CWsignal to generate a transmitted signal that is transmitted to acommunication link; adding, by a combiner of a reception path ofKramers-Kronig receiver, the CW signal with a received signal from thecommunication link to provide a combined signal; wherein the receivedsignal comprises a first sideband and a second sideband that is spacedapart from each other by a guard band; wherein the CW signal has afrequency that is included in the guard band; filtering, by a firstoptical filter of the reception path of Kramers-Kronig receiver, thecombined signal and output a first intermediate signal that comprisesthe CW signal and the first sideband; filtering, by a second opticalfilter of the reception path of Kramers-Kronig receiver, the combinedsignal and output a second intermediate signal that comprises the CWsignal and the second sideband; receiving by a first photodiode of thereception path of Kramers-Kronig receiver the first intermediate signaland outputting a first photocurrent that represents the firstintermediate signal; receiving, by a second photodiode of the receptionpath of Kramers-Kronig receiver, the second intermediate signal andoutputting a second photocurrent that represents the second intermediatesignal; performing an analog to digital conversion of the firstphotocurrent, by a first analog to digital converter of the receptionpath, to generate a digital representation of the first photocurrent;performing an analog to digital conversion of the second photocurrent,by a second analog to digital converter of the reception path, togenerate a digital representation of the second photocurrent; andprocessing, by a digital processor of the reception path ofKramers-Kronig receiver, the digital representation of the firstphotocurrent and the digital representation of the second photocurrentto provide a reconstructed received signal, wherein the processing isbased on a Kramers-Kronig relationship related to the received signal.10. A method, comprising: generating by a local oscillator a continuouswave (CW) signal; modulating, by a complex modulator, the CW signal togenerate a transmitted signal that is transmitted to a communicationlink; receiving a polarization-multiplexed signal by a polarizationdemultiplexing circuit of the reception path of Kramers-Kronig receiverand outputting a first polarization component of thepolarization-multiplexed signal and a second polarization component ofthe polarization-multiplexed signal; filtering, by a first opticalfilter of the reception path of Kramers-Kronig receiver, the firstpolarization component and outputting a first sideband of the firstpolarization component; filtering, by a second optical filter of thereception path of Kramers-Kronig receiver, the first polarizationcomponent and outputting a second sideband of the first polarizationcomponent; wherein the first and second sidebands of the firstpolarization component is spaced apart from each other by a guard band;combining, by a first combiner of the reception path of Kramers-Kronigreceiver, the CW signal with the first sideband of the firstpolarization component to provide a first combined signal; combining, bya second combiner of the reception path of Kramers-Kronig receiver theCW signal with the second sideband of the first polarization componentto provide a second combined signal; receiving, by a first photodiode ofthe reception path of Kramers-Kronig receiver, the first combined signaland outputting a first photocurrent that represents the first combinedsignal; receiving, by a second photodiode of the reception path ofKramers-Kronig receiver, the second combined signal and outputting asecond photocurrent that represents the second combined signal;converting, by a first analog to digital converter of the reception pathof Kramers-Kronig receiver, the first photocurrent to provide a digitalrepresentation of the first photocurrent; converting, by a second analogto digital converter of the reception path of Kramers-Kronig receiver,the second photocurrent to provide a digital representation of thesecond photocurrent; filtering, by a third optical filter of thereception path of Kramers-Kronig receiver, the second polarizationcomponent and outputting a first sideband of the second polarizationcomponent; filtering, by a fourth optical filter of the reception pathof Kramers-Kronig receiver, the second polarization component andoutputting a second sideband of the second polarization component;wherein the first and second sidebands of the second polarizationcomponent is spaced apart from each other by a guard band; combining, bya third combiner of the reception path of Kramers-Kronig receiver, theCW signal with the first sideband of the second polarization componentto provide a third combined signal; combining, by a fourth combiner ofthe reception path of Kramers-Kronig receiver the CW signal with thesecond sideband of the second polarization component to provide a fourthcombined signal; receiving, by a third photodiode of the reception pathof Kramers-Kronig receiver, the third combined signal and outputting athird photocurrent that represents the third combined signal; receiving,by a fourth photodiode of the reception path of Kramers-Kronig receiver,the fourth combined signal and outputting a fourth photocurrent thatrepresents the fourth combined signal; converting, by a third analog todigital converter of the reception path of Kramers-Kronig receiver, thethird photocurrent to provide a digital representation of the thirdphotocurrent; converting, by a fourth analog to digital converter of thereception path of Kramers-Kronig receiver, the fourth photocurrent toprovide a digital representation of the fourth photocurrent; andprocessing, by a digital processor of the reception path the digitalrepresentation of the first photocurrent, the second photocurrent, thethird photocurrent and the fourth photocurrent to provide areconstructed polarization-multiplexed signal, wherein the processing isbased on a Kramers-Kronig relationship related to thepolarization-multiplexed signal.